Construction of the Sparsest Maximally r-Robust Graphs

TL;DR

This paper investigates the minimal edge configurations needed for maximally r-robust graphs, crucial for resilient consensus in resource-constrained networks, and introduces two classes of such graphs with proven optimality.

Contribution

It provides tight lower bounds on edges for maximum robustness and introduces two classes of sparsest maximally r-robust graphs based on these bounds.

Findings

Derived necessary subgraph structures for maximum robustness.

Established tight lower bounds on the number of edges.

Proposed two classes of graphs achieving maximum robustness with minimal edges.

Abstract

In recent years, the notion of r-robustness for the communication graph of the network has been introduced to address the challenge of achieving consensus in the presence of misbehaving agents. Higher r-robustness typically implies higher tolerance to malicious information towards achieving resilient consensus, but it also implies more edges for the communication graph. This in turn conflicts with the need to minimize communication due to limited resources in real-world applications (e.g., multi-robot networks). In this paper, our contributions are twofold. (a) We provide the necessary subgraph structures and tight lower bounds on the number of edges required for graphs with a given number of nodes to achieve maximum robustness. (b) We then use the results of (a) to introduce two classes of graphs that maintain maximum robustness with the least number of edges. Our work is validated through a series of simulations.